Before I talk about addition and multiplication in mathematical Visual Poetry I would like to present the following two paintings by Giorgio De Chirico. These were created in the beginning years of the 20th century.
When I was visiting the inner harbor of Baltimore, Maryland I came across a most interesting tower. I later found the name to be "The Shot Tower". (Below)
As you can see, it is tall, cylindrical and has a little flag on the top of it. It reminded me of the towers I have seen in many Giorgio De Chirico paintings. I only included two painting here in this blog post but, there are many more that can be found in art history books.
So I got the idea to take it into Photoshop and turn the scene into a De Chirico-ish image.
I titled the piece: “THE QUESTION OF DE CHIRICO” and it poses the question: “Is the image on the right side of the piece equal to the ideas of Baltimore times De Chirico or is the image equal to the ideas of Baltimore plus De Chirico?
In my original post on this 'kogwork' I received a couple of responses that proved to me that it is an interesting question and the answer is not as esoteric as one might imagine. I will display and discuss the responses at the bottom of this blog entry.
I gave a lecture on Polyaesthetics and Mathematical Poetry last year at the Salk Institute and within the boundaries of my presentation I had a section that addressed this very issue. From that lecture I am going to borrow a few images to help illuminate this most interesting idea.
Let us think about the equation 3 + 4 = 7 and let us look at a pie chart to help illuminate our quest. When we add 3 and 4 together we can distinctly see the separate pieces within the pie as well as seeing the entire seven pieces. (Shown below)
The Bottom line is that it is easy to remove the 3 slices or the 4 slices from the mix of 7
Now let us think about the equation 3 x 4 = 12
When it comes to multiplication our task gets a little trickier tracking where the numbers 3 and 4 end up (visually). The difficulty is due to them get integrated into each other to produce the number 12. It is though they form an augmentation from which each other play a part in constructing. If we look at a pie chart again we can see that the 12 pieces can be viewed as 4 groups of 3 or we can view it as 3 groups of 4. Both numbers influence the whole in their own way.
Above we have 4 groups of 3 to yield the product of 12
Below we have 3 groups of 4 to yield the product of 12
So what we see is that the multiplier and the multiplicand both augment each other to produce the product.
So how does all of this relate to mathematical poetry? How can we multiply concepts or even images?
Let’s look at the next image titled "Americana Mathematics" and analyze its components.
We see an the popular American icon depicting a NASCAR racing machine added to an 8 ball from the game of pool to yield a strange vehicle that is part race car and part pool table. Here in this example as in our pie chart we can see the two concepts added in such a way that it would be easy to pull them apart and break them out of the whole. The two concepts can be clearly separated in addition however; in multiplication it is again trickier. Let’s look at 8 x 8 = 64 Here again we can refer back to our pie charts showing how the multiplier and multiplicand each augment the other idea to create a whole that possesses much more amplitude than the originating two concepts. Here our product is not a race car but a rocket ship that is obviously involved in some sort of pool game.
Now that we have the tools to understand the mechanics of this artwork we can then spend our time experiencing the interacting metaphors involved to come to our understanding of the signified.
I now want to post two responses to the original question of De Chirico from the blog entry on August 7, 2008.
The first being from the Math Poet TT.O.
The text in Blue is from TT.O. and the text in white is mine
My attempt at a solution to the difference in addition and multiplication in mathematical poetry is as follows:-----
As the difference in nomenclature suggests, the above problem of A+B=C and A*B=C may be a issue of semantics, and in the case of "mathematical poetry" the said equations NOT equal.
Consider One: A + B = C may mean let A abut B i.e. let image A physically touch image B, a kind of concatenation, a bringing together. Which would then go on to suggest that A + B = C1, and B + A = C2 since A + B ≠ B + A, and as their relative positions read from left-to-right would imply, the bringing together would result in an AB versus BA result. Notice that the collapse into a visual representation would suggest a kind of visual multiplication.
I want to add for any mathematicians that are reading this -- when he says A + B ≠ B + A we all realize that this is definitely not true in pure mathematics however, it is debatable within the context of mathematical poetry due to syntax having some bearing on the results. From my perspective the influence of syntax is minimal when performing addition, although, I am willing to listen to all arguments.
I will say that syntax is more important with multiplication. One can see the importance within the recent post I made called a+b+c does not equal c+b+a in this post our attention is brought to a problem with the order in which one experiences a phenomena. The author titled his observation a+b+c does not equal c+b+a however I believe that he should have realized what he was performing was multiplication not addition.
Consider Two: A * B = C may depend on how it is read i.e. a issue of semantics (again) i.e. the number to be multiplied is called the "multiplicand", while the number of multiples is called the "multiplier". Perhaps this is better seen in the following equation A ( B + C ) = D. Here, the multiplier is A while the multiplicand is (B + C). The semantics of the equation would then suggest that ( B + C ) A ≠ A (B + C) in mathematical poetry, since it would depend on which was the multiplicand and which the multiplier, and in what order they were being taken to be (or read) i.e. what was to be infused by what, or what was to be increased by what i.e. a kind of what is being "acted on" (passive) and what active.
Here TT.O. has provided a good argument to warrant attention being paid to the syntax of the equation within the context of mathematical poetry. However, there could be an argument that within the realm of pure math syntax makes no difference and therefore the poet needs to create his/her metaphor to reflect this mathematical truth. In other words make the product reflect an equal amount of the conceptual essence of the multiplier and multiplicand.
From where I stand, in the equation A + B = C, A cannot infuse into B (or visa versa), but can only stand-by it. Multiplication, in the equation A * B = C, on the other hand (to carry on the metaphor) "impregnates" B but not visa versa.
I don't understand your poem properly, because I don't understand the basic essence of De Chirico's work (i.e. a specific painting???) or who or what Baltimore is i.e. a City? An Artist? An attitude?
However, I would suggest that Baltimore × De Chirico is different from De Chirico × Baltimore and different to Baltimore + De Chirico, and De Chirico + Baltimore, and that we should be mindful of it in our equation making.
TT.O.
I want to thank TT.O. for commenting on “The Question of De Chirico” and I must ask forgiveness for not explaining that the image is one of my photographs of a tower that resides on the inner harbor landscape in downtown Baltimore, Maryland USA. I modified the image to be in the style of the twentieth century painter Giorgio De Chirico. (See Google)
Here is a few excerpts from a response from Todd Smith:
Here's my take on it: The painting on the right seems to fit the style of the painter Giorgio de Chirico, so I assume that it is his work. If this is the case, I would vote for the equation: de Chirico (Baltimore) i.e., multiplication.
Multiplication implies a combination (almost a mixing of two elements) and it generates something more than the sum of the two entities being combined.
I would suggest that a snap shot of de Chirico with Baltimore in the background to be represented by the equation de Chirico + Baltimore.
But a work of art produced by de Chirico in which Baltimore is featured would mean multiplication to me. The painting is as much de Chirico as it is Baltimore. The two are inextricably intertwined.
Multiplication seems to be a more complex combination than addition to me. Two spools of thread might be added together when placed in a shopping bag, but they would be multiplied together if they were woven into a shirt.
Here is an image (above) which illustrates Todd's idea of a mathematical weave between two axes. The image is titled "Distance" and it uses the distace equation:
Distance = velocity multiplied by time.
Also, addition seems to be one-dimensional, while multiplication seems to create two dimensions. Addition happens along the number line, while multiplication can be graphed along the x and y axis.
They say you can't add apples and oranges. In addition you have to find a common denominator before you can add. This implies the number line again. As soon as two things are on the same dimension they can be added. For example, de Chirico and Baltimore are both physical things and so they can both be photographed together and said to be "added together" in the picture.
But with multiplication there is less restriction. You don't need a common denominator to multiply two things. The combination creates something new that is not merely more quantity of a common denominator. In pure mathematics 3 x 4 creates a rectangle of area 12. Before there were only lines (one dimension), after multiplication there is area (two dimensions). New space is created. In the example of de Chirico, Baltimore x de Chirico created a new vision of Baltimore colored by de Chirico's own inspiration. No one had seen Baltimore in quite the same way. It is as if a new dimension was opened when these two were combined.
Well, I didn't plan to write this much, but it's fun to think about.
Thanks,
Todd
I also want to thank Todd Smith for his wonderful comments as well. I think the point that we all would like to assert is that this idea of adding and multiplying images (or concepts) is easy to understand. I would love to see more from everyone out there.
Thanks.
Kaz
This blog entry is to share a few moments and images of the AMS and MAA Joint Meeting held this year in Washington D.C. The first lonely image is a view from my hotel room bed as I tried to get to sleep by counting the sheep jumping over the Washington Monument. The weather for the most part was cold and rainy and made my three-quarter mile walk to the conference a little trying at times. However the last day was nice and offered the image below which was shot from my hotel room window as the sun was rising in the east.
The shot below is a 180 degree panoramic view of 21 images sewn together to provide an overview of the entire show of mathart. The show is always modestly done due to its modest budget (relative to art galleries) but it is always done well and the people who work on it are wonderful enthusiastic individuals who feel mathart can make a difference. They even gave out prizes this year.
The shot below is of the past president of the Mathematical Association of America, Joseph Gallian as he was browsing the show.
Speaking of browsing the show … The next image (below) is of our friend Ivars Peterson who writes about mathart for many publications including Science News. 
From left to right: Reza Sarhangi the nucleus of the Bridges Conference on mathematical connections in art music and architecture , Annette Emerson, the Public Relations Officer of the AMS and Anne Burns, one of the Judges for the mathart exhibit and webmaster for the mathart exhibit webpage.
The next image is of Robert Fathauer who curates the mathart show each year and also owns and operated Tessellations a company devoted to selling objects that inspire the math aesthetic.
Here is Nat Friedman and one of his knots displaying a minimum surface by spanning a soap bubble film across the knot. Very interesting and simple stuff showing complex concepts.
The next image is of Reza Sarhangi with Arthur Benjamin who happens to spend a lot of time on stage racing calculators … I have seen him in action and yes, he can calculate in his brain faster than you can calculate on your hand calculator. Check out this video. 
JoAnne Growney and Sarah Glaz recently edited an anthology of mathematical love poems titled “Strange Attractors”. The book was published by AK Peters and can be seen in the above photo at the bottom right of the image. Also in the photo are Klaus Peters (left) and his lovely wife Alice (Thus AK). The conference also provided a poetry reading session to deliver poems from the book. The event was organized by co-editor JoAnne and you can see the crowd gathering for the reading in the image below.
The image above is of me delivering my poem "Prometheus's Epistle to Job"
Karl Kempton
